Preparation

• Strip the tarragon leaves from the stems, finely chop the leaves to yield about 2 tsp., and set aside.
• In an 8-inch skillet, melt 1 Tbs. of the butter over medium-low heat. Add the shallot and tarragon stems, and cook, stirring frequently, until the shallot is translucent, about 2 minutes. Add the wine, vinegar, and 1 Tbs. water, increase the heat to medium, and cook until the liquid is mostly evaporated, about 3 minutes. Transfer to a small bowl, remove and discard the tarragon stems, and let cool completely. Add the remaining 3 Tbs. butter, the chopped tarragon, 1/4 tsp. salt, and 1/8 tsp. pepper, and mix with a fork until well combined. Set aside until ready to use.
• Prepare a medium-high (400°F to 475°F) gas or charcoal grill fire. Put the meat and asparagus on separate sides of a large rimmed baking sheet (or similar). Coat with the olive oil, and season generously with salt and pepper. Grill the meat, turning every 2 minutes, until cooked to your liking, 6 to 8 minutes for medium (140°F). Transfer to a cutting board, cover loosely with foil, and let rest. Meanwhile, grill the asparagus, turning after 1 minute, until charred and tender, 2 to 3 minutes. Transfer to a serving platter. Thinly slice the meat against the grain, arrange on the platter with the asparagus, top both with dollops of the butter, and serve.

Make Ahead Tips

The butter may be made up to 1 week ahead; refrigerate and return to room temperature before using.